This is a topic in classical algebraic K-Theory. I will recall
definitions of elementary linear group, elementary symplectic group,
linear transvection group, and symplectic transvection group. These groups
have natural action on the set of unimodular elements. I will briefly
discuss how bijections between orbit spaces of unimodular elements under
different group actions are established. Finally, I will talk about an
application of these results, namely improving injective stability bound
for the K1 group.